The abc Conjecture and Fermat's Last Theorem

An article appeared in Scientific American's website on April 3rd 2020 titled Mathematical Proof That Rocked Number Theory Will Be Published and which began:

Figure 1: Shinichi Mochizuki

After an eight-year struggle, embattled Japanese mathematician Shinichi Mochizuki has finally received some validation. His 600-page proof of the abc conjecture, one of the biggest open problems in number theory, has been accepted for publication. 

Acceptance of the work in Publications of the Research Institute for Mathematical Sciences (RIMS)—a journal of which Mochizuki is chief editor, published by the institute where he works at Kyoto University—is the latest development in a long and acrimonious controversy over the mathematicians’ proof. 

Two other RIMS mathematicians, Masaki Kashiwara and Akio Tamagawa, announced in Japanese the publication at a 3 April press conference in Kyoto. The paper “will have a big impact”, said Kashiwara. When asked how Mochizuki reacted to news of the paper’s acceptance, Kashiwara said, “I think he was relieved.” 

Mochizuki, who has denied requests for interviews over the years, did not appear and did not make himself available to reporters. 

Eight years ago, Mochizuki posted four massive papers online, claiming to have solved the abc conjecture. The work baffled mathematicians, who spent years trying to understand it. Then, in 2018, two highly respected mathematicians said they were confident that they had found a flaw in Mochizuki’s proof—something many saw as death blow to his claims. 

The latest announcement seems unlikely to move many researchers over to Mochizuki’s camp. “I think it is safe to say that there has not been much change in the community opinion since 2018,” says Kiran Kedlaya, a number theorist at the University of California, San Diego, who was among the experts who put considerable effort over several years trying to verify the proof. Another mathematician, Edward Frenkel of the University of California, Berkeley, says, “I will withhold my judgment on the publication of this work until it actually happens, as new information might emerge.”

The article goes on to describe the abc conjecture in these terms:

The ‘abc conjecture’, the problem Mochizuki claims to have solved, expresses a profound link between the addition and multiplication of integer numbers. Any integer can be factored into prime numbers, its ‘divisors’: for example, 60 = 5 x 3 x 2 x 2. The conjecture roughly states that if a lot of small primes divide two numbers \(a\) and \(b\), then only a few, large ones divide their sum, \(c\). 

Let's look at a different definition now, taken from Wikipedia:

Take three positive integers, \(a\), \(b\) and \(c\) (hence the name) that are relatively prime and satisfy \(a + b = c\). If \(d\) denotes the product of the distinct prime factors of \(a \times b \times c\), the conjecture essentially states that \(d\) is usually not much smaller than \(c\). In other words: if \(a\) and \(b\) are composed from large powers of primes, then \(c\) is usually not divisible by large powers of primes.

What's highlighted in red is essentially saying the same thing and provides an approach to proving Fermat's Last Theorem. To see this, let's look at \(a=13^{22}\) and \(b=11^{22}\) as an example. We know that Fermat's Last Theorem hopes to find a value for \(n>2\) such that:$$a^n+b^n=c^n \text{ for integer values of }a,b,c$$However, the \(abc\) conjecture indicates that for \(13^{22}+11^{22}\) this is highly unlikely. In fact, we have:$$13^{22}+11^{22}=2 \times 5 \times 29 \times 44617 \times 955769 \times 266300690657$$Here is another example:$$101^{19}+197^{19}=2 \times 149 \times 229 \times 1483 \times 110398608667213673 \times 3521300862956110103$$So from those two examples, one gets a glimpse of how a generalised approach using the \(abc\) conjecture might be used to provide a proof of Fermat's Last Theorem.

Of course, neither of the previous two definitions is a formal definition of the conjecture. We'll get to that shortly. In the meantime, here is a Numberphile video that appeared soon after Mochizuki's paper appeared:

In the video, the conjecture is described as follows where rad is the product of the distinct prime factors:

If \(a\) and \(b\) are coprime integers and \(a\)+\(b\)=\(c\) then (in general):$$ \text{rad}(abc)^k>c$$If \(k=1\) there are infinitely many exceptions.

If \(k>1\) there are finitely many exceptions.

He quotes \(3 + 125 =128\) as an example of an exception when \(k=1\). In this case, we have \( \text{rad}(3, 125, 128) = 3 \times 5 \times 2 = 30 < 128\). For the examples that I used earlier however, the opposite is the case. Figure 1 shows a SageMathCell screenshot for \(13^{22}+11^{22}\), with permalink included:

Figure 1: permalink

The definitions can get more technical than those quoted earlier but that's probably enough for now.